Singular boundary method in free vibration analysis of compound liquid-filled shells

1. 42nd International Conference on Boundary Elements and other Mesh Reduction Methods

SINGULAR BOUNDARY METHOD
IN FREE VIBRATION ANALYSIS
OF COMPOUND LIQUID-FILLED SHELLS
VASYL GNITKO, KYRYL DEGTYARIOV,
ARTEM KARAIEV,
ELENA STRELNIKOVA
V. N. Karazin Kharkiv National University
A. Podgorny Institute of Mechanical Engineering Problems
UKRAINE
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2. A.N. Podgorny Institute of Mechanical Engineering Problems, National Academy of Sciences, Ukraine

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3. CONTENTS

• Introduction and problem statement
• Mode superposition method for coupled dynamic
problems
• Systems of the boundary integral equations and
some remarks about their numerical implementation
• Some numerical results
• Conclusion
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4. A. Podgorny Institute of Mechanical Engineering Problems

• The A.N.Podgorny Institute for Mechanical Engineering Problems
of the National Academy of Sciences of Ukraine
• (IPMash NAS of Ukraine) is a renown research centre in
power and mechanical engineering.
IPMash has 5 research departments with a staff of 346 specialists (133
research workers, including one Academician and five Corresponding
Members of NAS of Ukraine; and 32 Doctors and 77 Candidates of
Science). The Institute also has a special Design-and-Engineering Bureau,
and a pilot production facility.
• Key research areas
optimisation of processes in power machinery
energy saving technologies
predicting the reliability, dynamic strength and life of power equipment;
simulation and computer technologies in power machine building
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5. LIQUID FILLED SHELLS

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6. PROBLEM STATEMENT

) pn
L(U) M(U
0
p l
gz p0
t
• with the next set of boundary conditions relative to
w
n S1 t
n
S0
;
t
gz 0
t
s0
• w=(U,n)
• fixation conditions of the shell relative to U
• Initial conditions
x, y,0 H ; x, y,0 0
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7. BOUNDARY CONDITIONS ON ELASTIC AND RIGID SURFACES

U , n
grad n s
; grad n s2 0
t
1
HARMONIC VIBRATIONS
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8. MODE DECOMPOSITION METHOD FOR COUPLED DYNAMIC PROBLEMS

• Displacements are linear combination of
structure natural modes without liquid
u
N
c u
k
k
k 1
uk are the normal modes of vibrations of the empty shell.
the first system of basic functions
Representation for velocity potential
1 2
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9. BOUNDARY VALUE PROBLEM FOR POTENTIAL

1
POTENTIAL
1 DEFINES ELASTIC WALL VIBRATIONS second
system of basic functions
2
1 0
1 w
, P S1
n
t
(1 x, y, z, t )
1
0 , P S0
t
m
( x, y, z )c ( t )
1k
k
k 1
• Boundary value problems for functions 1k
2
1k 0
1k
wk
n S 1
1k 0 , P S 0
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10. Boundary value problem for velocity potential third system of basic functions

Boundary value problem for velocity
potential 2
third system of basic functions
2 2 0
2
2
0 , P S1
, P S0
n
n
2
g a s t x 0
t
s0
• representation for velocity potential
2
• harmonic vibrations of liquid in rigid shell
2( x, y, z, t ) ei t ( x, y, z )
2
• 0
0 , P S1
n
2
, P S0
n g
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11. SPRING- PENDULUM ANALOGY

(1 x, y, z, t )
m
( x, y, z )c ( t )
1k
k
k 1
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12. Three systems of basic functions

• At first we obtain the natural modes and frequencies of
structure without liquid – the first system of basic functions
• Second, we represent the velocity potential
• as a sum 1 2 and for each component consider the
corresponding boundary value problem for Laplace equation.
• The potential 1 corresponds to the problem of elastic
structure vibrations with the liquid but without including the
force of gravity - the second system of basic functions
• The potential 2 corresponds to the problem of rigid structure
vibrations with the liquid including the force of gravity • the third system of basic functions
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13. EIGENVALUE PROBLEM

FOR HARMONIC VIBRATIONS WE HAVE
ck t Ck exp(i t );
d k t Dk exp(i t );
aS t 0
We use the direct BEM formulation
1
1
2 P0
dS
dS
n P P0
n P P0
S
S
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14. FIRST AND SECOND BASIC FUNCTIONS

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15. VIBRATIONS OF RECTANGULAR PLATES

h, м
0,00215
0,003
0,005
air.
experiment/
calculation
Reduction
coefficient
194/197
290/280
413/458
1,57
1,27
1,23
1
Liquid
Reduction
coefficient
Liquid
experiment
BEM
112
182
360
1
1,75
1,53
1,27
106
179
305
Reduction
coefficient
1,82
1,62
1,21
MODES OF VIBRATIONS
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16. VIBRATIONS OF SECTORIAL PLATES

Number of
frequencies, Hz
frequency
liquid
air
1
2
3
4
experiment
calculation
experiment
calculation
402
416
514
714
398.2
425.0
549.7
791.0
159
277
420
161
242
300
460
MODES OF VIBRATIONS
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17. VIBRATIOBS OF FRANSIS TURBINE

Without added liquid masses
Number of
frequency
1
2,3
4,5
6,7
8
FEM
Leningrad metal plant
Experiment
TURBOATOM
experiment
33.05
35.80
42.90
71.90
80.70
33.68
37.13
39.87
57.89
80.44
33.87
35.93
63.06
66.99
83.83
With added liquid masses
Number of
frequency
1
2,3
4,5
6,7
8
FEM
24.00
29.20
31.50
37.00
52.50
Leningrad metal
plant
Experiment
22.5
28.5
31.1
33.3
TURBOATOM
experiment
21.6
28.5
32.7
37.2
40.2
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18. FRANSIS TURBINE MODES OF VIBRATIONS

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19. Free vibrations of liquid in rigid shells. Boundary value problem. Third system of basic functions.

2 0
( x, y, z, t ) ei t ( x, y, z )
0 , P S1
n
0
, P S0
z
0 , P S1
n
g 0 , P S0
2
Free vibrations
2
, P S0
n
g
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20. Singular boundary method

THIRD GREEN’S IDENTITY
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21. ONE-DIMENSIONAL BOUNDARY ELEMENTS

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22. Singular boundary method

• We divide boundary into N boundary elements; m
elements belong to the free surface and others N-m
belong to the shell walls. At each boundary element
we select the collocation point